Overview

This concise textbook, fashioned along the syllabus for master’s and Ph.D. programmes, covers basic results on discrete-time martingales and applications. It includes additional interesting and useful topics, providing the ability to move beyond. Adequate details are provided with exercises within the text and at the end of chapters. Basic results include Doob’s optional sampling theorem, Wald identities, Doob’s maximal inequality, upcrossing lemma, time-reversed martingales, a variety of convergence results and a limited discussion of the Burkholder inequalities. Applications include the 0-1 laws of Kolmogorov and Hewitt–Savage, the strong laws for U-statistics and exchangeable sequences, De Finetti’s theorem for exchangeable sequences and Kakutani’s theorem for product martingales. A simple central limit theorem for martingales is proven and applied to a basic urn model, the trace of a random matrix and Markov chains. Additional topics include forward martingale representation for U-statistics, conditional Borel–Cantelli lemma, Azuma–Hoeffding inequality, conditional three series theorem, strong law for martingales and the Kesten–Stigum theorem for a simple branching process. The prerequisite for this course is a first course in measure theoretic probability. The book recollects its essential concepts and results, mostly without proof, but full details have been provided for the Radon–Nikodym theorem and the concept of conditional expectation.

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9789819744718

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Arup Bose is a professor at the Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Kolkata, West Bengal, India. He has research contributions in statistics, probability, economics and econometrics. A recipient of the P.C. Mahalanobis International Prize in Statistics, S.S. Bhatnagar Prize, the C.R. Rao Award and holds a J.C. Bose Fellowship, he is a fellow of the Institute of Mathematical Statistics (USA) and all three Indian national science academies. He has authored five books: Random Matrices and Non-commutative Probability, Patterned Random Matrices, Large Covariance and Autocovariance Matrices (with Monika Bhattacharjee), Random Circulant Matrices (with Koushik Saha) and U-Statistics, M_m-Estimators and Resampling (with Snigdhansu Chatterjee). Arijit Chakrabarty is has been an associate professor at the Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Kolkata, West Bengal, India, since 2016. Earlier, he was at the Delhi Centre of the same institute. He obtained his B. Stat. and M. Stat. degrees from the Indian Statistical Institute, Kolkata, and his Ph.D. from Cornell University, USA. His research area is probability theory.   Rajat Subhra Hazra has been an associate professor of mathematics at Leiden University, the Netherlands, since 2021. He also worked as a faculty at the Indian Statistical Institute, Kolkata, India, from 2014 to 2021. A recipient of the S.S. Bhatnagar Prize, he is a fellow of the Indian National Science Academy, New Delhi. His research area is probability theory.

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